Smallest stable black-holes

CraigD · 07-08-2009

Quote:

Originally Posted by Hasanuddin

As to the primary question of the smallest size that a black-hole can be and exist stably.

Quote:

Originally Posted by CraigD

For example, a 2e5 kg black hole has a lifetime of about 1 second, a 7e5 kg one about 1 year, and a 1e11 kg one about 1.4e10 s, about the current age of the universe according to Big Bang model. By way of comparison, 1e11 kg is just a bit more than the mass of a large artificial structure like the Three Gorges Dam.

Okay, let me just clarify… you are talking about only the evaporating side of the equation… correct?

Yes. The most simple evaporation time equation is just an integral of the power – the rate of Hawking radiation of mass/energy, given by

$P = \frac{k_P}{M^2}$ ,
where $k_P=\frac{\hbar c^6}{15360 \pi G^2}$ ,
and $M$ is the black hole’s mass

– of a black hole as its mass, and consequently the radius and surface area of its event horizon, which determine that power, changes, decreased by the outflowing Hawking radiation. It doesn’t include inflowing radiation.

Note that it’s common and useful for power to be expressed as an equivalent temperature of a black body of given surface area, so it’s common to see the above expressed in, for example, degrees Kelvin (K), rather than watts (W) or other common units of mechanical power.

Note that the term “radiation” isn’t limited to photons, but describes anything that carries mass/energy from one body to another – and that “body” can refer to a well-defined volume such as within the event horizon of a specific black hole, or all of space not within that specific black hole.

Quote:

Originally Posted by Hasanuddin

What about the other side of the dynamic? Consumption. Does a black-hole consume more when the density of compactable material is high? Deductively the answer must be, “Yes.”

This answer has to be approached carefully, taking special care not to confuse mass and density (mass/volume).

As noted above, and in the linked wikipedia article, incoming and outgoing radiation must be considered for a full description of the mass of a black hole over time. Density, however, other that a critical value, isn’t a term in this description.

Assuming a small rate of rotation and nearly neutral net charge, the event horizon of a black hole is nearly spherical, its radius determined by its mass only, by the very strait-forward Schwarzschild radius,

$r_s = k M$ ,
where $k_s = \frac{2 G}{c^2}$

Density is important only in that it must be sufficiently large, on average, that the mass/energy responsible for the black hole’s gravity is within the volume defined by its event horizon. Because the Schwarzschild radius is proportional to M, and volume is proportional to $r_s^3$ , the average density of a black hole can be arbitrarily small. For example, at about 130,000,000 solar masses, the average density of a non-rotating black hole is about equal to that of water, and rather curiously, if the visible universe were a single massive black hole, its required density would be very close to the very hard vacuum given by various predicted values of its total mass.

It’s unlikely, according to sensible physics, that the mass within a black hole is anything close to evenly distributed, so average density of its entire volume is almost certainly very different than the average density of various sub-volumes, the most accepted guess being that most of the volume is near vacuum, with a tremendously dense core - possibly an infinitely dense singularity, though, as the saying goes, nature – especially when viewed with the formalism of quantum physics - abhors infinities, so my guess is for “tremendous” of “infinitely”. However, the question of densities within the event horizon of a black hole isn’t important to the physics of anything outside of it, or, in theory, very knowable.

Quote:

Originally Posted by Hasanuddin

If a black-hole is in a pure vacuum then nothing could be consumed.

Here, we must be careful to agree on the meaning of “pure vacuum”.

The most common meaning is a volume that contains no fermionic matter (atoms or free nuclei or electrons, etc), but may contain bosonic matter (photons, etc).

However, because typical black holes Hawking radiate with such low power, a small influx of photons is sufficient to more than equal the outgoing radiation. This is what I mean by my statement (the numbers lifted directly from the wikipedia article)

Quote:

Originally Posted by CraigD

In principle, a small black hole could be stable – that is, be at equilibrium, neither gaining nor losing mass – if the power of its infalling matter and radiation equals that of its Hawking radiation. For the cosmic background radiation – which all objects are more or less guaranteed to receive – a black hole of about $4 \times 10^{22}$ kg – about the mass of Earth’s moon has about this equilibrium.

So a black hole with the mass of the moon will, barring some bizarre “shadowing” phenomena, gain mass just via absorbing the CMBR. One smaller might still be stable or gain mass, if the influx of photons was greater – say from a nearby bright star – or if more than a small amount of matter – such as in typical near-stellar space – fell into it.

However, a black hole much smaller than this will, barring an extraordinary influx of bosons and/or fermions, Hawking radiate much more than it absorbs, so lose mass at an ever increasing rate. The main consequence of this is that, according the theory, small black holes, which can in principle be formed by such small-scale phenomena as man-made particle accelerators, will exist for only very short durations – a reassuring prediction, as it reduces our worries of swarms of tiny black holes from cosmic sources, or created by high-energy physics experiments, devouring our Sun or planets.

Quote:

Originally Posted by Hasanuddin

Perhaps I am misreading you, but it appears that you are saying that equilibrium for a black-hole only assimilating/accreting the energy from the CMB, but nothing else, will be stable at a mass of 4e22 kg. Am I reading you correctly? So, a smaller, yet still stable black-hole could be achieved under conditions where more energy/mass are being accreted. Is that correct?

Yes, as I hope the preceding explains more completely.

Quote:

Originally Posted by Hasanuddin

Quote:

Originally Posted by CraigD

(infalling matter couldn’t be used to stabilize an arbitrarily small black hole, because the exclusion principle limits the amount of fermionic mater that can occupy a given volume of space.)

The problem is, following the link provided offers no evidence to support the notion that infalling energy would be interchangeable infalling mass to achieve black-hole stability. After all doesn’t the famous equation E=mc2 suggest an interchangeability between mass and energy? Nowhere on the link discussing the Pauli Exclusion Principle is it suggested that E=mc2 does not apply.

What I’m trying to explain with this reference is that, although under usual conditions, streams of matter (fermionic) are much higher power than streams of photons (bosons), fermionic matter streams have an upper limit to the power they can supply to a surface of given area, while photon streams don’t. This is because the Pauli Exclusion principle – AKA Fermi-Dirac statistics – limits how many fermions can occupy a given volume, while no such limit applies to bosons – or, to put it simply, the number of photons that can be contained in a given volume is unlimited.

What this all gets at concerning the minimum size of a black hole, is that while no know natural phenomena can prevent a black hole much smaller than $10^{22}$ kg from losing mass, fairly quickly “evaporating” completely, it’s conceivable that one might artificially sustain a very small black hole by, put simply, shining a very bright light on it.

Using the Hawking radiation power and Schwarzschild radius formulae above, we can calculate their constants for standard units,
$k_P = \frac{\hbar c^6}{15360 \pi G^2} \dot= 7.1 \times 10^{32} \,\mbox{W} \cdot \mbox{kg}^2$
$k_s = \frac{2 G}{c^2} \dot= 1.5 \times 10^{-27} \,\mbox{m/kg}$
, chart the various masses M, Schwarzschild radii r, Hawking radiation power P, and evaporation time t:

Code:

M (kg)  r (m)    P (W)    t (s)   (y)      Comments
1.0e0   1.5e-27  7.1e32   8.4e-17 2.7e-24
1.4e3   2.1e-24  3.6e26   2.3e-7  7.3e-15  Power of the Sun
1.0e4   1.5e-23  7.1e24   8.4e-5  2.7e-12
1.0e5   1.5e-22  7.1e22   8.4e-2  2.7e-9
5.0e5   7.4e-22  2.9e21   1.0e1   3.2e-7   500 tons, 1 second
1.0e6   1.5e-21  7.1e20   8.4e1   2.7e-6
1.0e7   1.5e-20  7.1e18   8.4e4   2.7e-3
1.0e8   1.5e-19  7.1e16   8.4e7   2.7e0
8.0e8   1.2e-18  1.1e15   4.3e10  1.4e3    Most powerful laser
1.0e9   1.5e-18  7.1e14   8.4e10  2.7e3
1.0e10  1.5e-17  7.1e12   8.4e13  2.7e6
1.5e10  2.2e-17  3.2e12   2.8e14  8.9e6    Power of human civilization
1.0e11  1.5e-16  7.1e10   8.4e16  2.7e9    About 1/4th age of the universe
1.0e12  1.5e-15  7.1e8    8.4e19  2.7e12   Proton’s radius
1.0e13  1.5e-14  7.1e6    8.4e22  2.7e15   Uranium nucleus’s radius
3.6e16  5.3e-11  5.5e-1   3.9e33  1.2e26   Hydrogen atom’s radius
4.5e22  6.7e-5   3.5e-13  7.7e51  2.4e44   Moon’s mass, hair’s radius
3.0e24  4.5e-3   7.9e-17  2.3e57  7.3e49   Small ball bearing’s radius
6.0e24  8.9e-3   2.0e-17  1.8e58  5.7e50   Earth’s mass
2.0e30  3.0e3    1.8e-28  6.7e74  2.1e67   Sun’s mass

, and consider what would be required to artificially stabilize various small black holes.

From the first rows of the table, we can see that until we get an initial black hole with a mass of over about a million tons (1.0e9 kg), the power necessary to sustain it is prohibitively high for a civilization of our technological level. At these initial masses, the black hole is fairly long-lived by human standards - about 2700 (2.7e3) years - so for practical purposes, there’s not much point in bothering to sustain it.

Although black holes of these masses are tiny, with event horizons about 1/1000th the size of a proton (1.5e-18 m), their close-in gravitational fields are very strong – for a 1e9 kg black hole, the acceleration of gravity exceeds Earth’s surface’s at a distance of about 8 cm (0.08 m). This makes them potential doomsday objects, as, per $E=mc^2$ , their daunting 7.1e14 W sustaining power requirements translate to only about 0.47 kg/minute (7.8e-3 kg/s) of matter.

Finally, there’s the engineering problem of how to make (or find) a sub-subatomic size, million ton black hole in the first place. If exploding stars are any indication, you’d need system masses several powers of 10 larger than the resulting black hole, and power many times greater than whole stars. We might hold out hope that in the many varieties of supernova’s in our galactic neighborhood, if we can manage to build spacecraft to visit some of these events, we might get lucky and find a small black hole, but quantum mechanics – among them those pesky Fermi-Dirac statistics again - predicts this is impossible, and observations showing the lack of strong x-ray emitters from sub-typical black hole size supernovae support these theoretical predictions.

What we’re left with, as best I can surmise, is the prospect that very small (less than 1000 kg) black holes with very short natural evaporation times might be possible to artificially produce and sustain by a technological civilization that could project focus radiation with the power of many sun’s on tiny targets. Otherwise, it appears the usual minimum mass prediction – about 1.4 solar masses – applies, resulting in black holes that won’t evaporate until the entire universe becomes nearly completely dark and on the order of $10^{100}$ years pass – the end of the “black hole era” predicted by some cosmological models.

Jay-qu · 07-08-2009

Thanks Craig for that informative post

So the Hawking radiation that leaves a black hole will be thermal? will the black hole act like a black body?

I am just wondering about the prospect of using a micro black hole as a power source, though likely the amount of energy that needs to be put in initially would outweigh its use as an output..

Hasanuddin · 07-09-2009

Ja-qu,

I will respond to you on the proper thread that regards AMBH: http://hypography.com/forums/physics...sequences.html

Hasanuddin · 07-09-2009

Wow, honestly, thank you for the informative and well-thought out post. There was a part, however, that you misinterpreted the line of my inquiry and spent much time tangent.

Quote:

Originally Posted by Hasanuddin

What about the other side of the dynamic? Consumption. Does a black-hole consume more when the density of compactable material is high? Deductively the answer must be, “Yes.”

I still adhere to the assertion. I was not referring to the density of the black-hole, I was referring to the density of the matrix of the region of space surrounding the baby black-hole. Let me lay it out a little more syllogistically:

1: If there is no matter/photons/other in the vicinity of the bitty black-hole, then the MBH will receive nothing.

2: If there is matter/photons/other within the vicinity of the MBH, it will accrete/absorb.
2b: If it absorbs energy/mass, then it gains energy/mass
2c:All gains in energy/mass, offset possible losses through proposed evaporation.
2d: Therefore, a lower total initial mass is needed by the MBH to sustain stable equilibrium between evaporation and consumption.
2e: Therefore, shouldn’t areas of highest neighborhood density of accretable mass/energy also be the areas where a black-hole could exist in the smallest of sizes indefinitely?

So the question is, what size would that be? I mean chose the extreme: hot Pb (lead). Suppose we has a giant block of very hot lead under high pressure. Within those conditions, how small is the smallest size that a black-hole can be?

For the sake of this question, please put the question of photonic energy aside. The last post seemed to detail the ability of light to sustain black-holes for a very long time. Cool. However, I am left wondering the role of accreted matter in sustaining an MBH. So, the question of light aside, how small is the smallest black-hole to be under mass-dense and high heat conditions?

The closest we got in approaching the true question I am trying to ask is when:

Quote:

Originally Posted by CraigD

although under usual conditions, streams of matter (fermionic) are much higher power than streams of photons (bosons), fermionic matter streams have an upper limit to the power they can supply to a surface of given area,

But you didn’t follow the natural thought progression. Please try. Under conditions of high fermionic surrounding matrix density (not the MBH’s density), under high pressure and high energy, what is the smallest size for an MBH to be able to stably persist?

modest · 07-09-2009

Quote:

Originally Posted by Jay-qu

I am just wondering about the prospect of using a micro black hole as a power source

Brilliant! I've never heard or considered that. It seems like it would be the perfect power source. You put in matter and it's converted to pure thermal radiation. It'd be the perfect rocket engine

Of course, attaching it to the rocket might present a problem

~modest

P.S. I've attached Craig's table with formulas in Excel if anybody wants to play with the #s

HydrogenBond · 07-09-2009

Do smaller black holes, such as the minimum mass of 3.8 solar masses, have the same center point reference as huge black holes? The larger black hole will have more mass, but once blackholes reach a point center, is that reference the same for both?

Jay-qu · 07-09-2009

Quote:

Originally Posted by Hasanuddin

The closest we got in approaching the true question I am trying to ask is when:
But you didn’t follow the natural thought progression. Please try. Under conditions of high fermionic surrounding matrix density (not the MBH’s density), under high pressure and high energy, what is the smallest size for an MBH to be able to stably persist?

I think if you paid a bit more attention and read his post thoroughly you would have got your answer.

To keep a very small black hole stable you must put in as much energy/mass as it is spitting out - simply an equilibrium of input/outputs.

The smaller a black hole gets the greater the rate at which it spits out hawking radiation. So you have to cram more mass back into the black hole at an much higher rate. There is a limit where the black hole becomes so small that you cant cram in mass fast enough due to limits imposed by the Pauli exclusion principle. This limit only occurs for fermions, so you could conceivably continue cramming energy/mass in with bosons.

The logical conclusion of this is, you can make an arbitrarily small black hole, so long as you have to means to continually pump in a very high flux of bosons.

Eric · 07-10-2009

There's a couple of important issues that have been left out of this.

One is that duration for micro black holes under large extradimensional string theory cannot be obtained by using the classical formulae for power and duration of black holes given by CraigD (this page). Secondly, even the type of large extra dimensional string theory can effect the result. This latter occurs where the extra dimension calculation takes into account the effect of gravity upon the radiated particles. When this is done it is called the 'microcanonical interpretation', where - for a given black hole mass - the energy, not temperature, is held fixed. I fail to see any reason why not taking this 'backreaction' effect into account can be regarded as valid.

The fact that within extra dimensions the duration of micro black holes is presented so much by CERN and elsewhere as extremely rapid, is a terrible disaster..

When the 'Randall-Sundrum' theory was considered in the paper of 2002 (also published in the 'International Journal of Modern Physics') http://arxiv.org/abs/hep-th/011025 by applying this alternative interpretation, the duration of a minimum possible 1TeV[/c^2] black hole in isolation is given at upto 30years. Though this theory is also the basis for the 2009 paperhttp://arxiv.org/abs/0901.2948, given in Hasanuddin's first thread post, parameters used in the earlier are not calculated in the new. Yet in the new, the other parameters are still described as 'Another possibility..'. This later neglect of calculation for risk evaluation demonstrates:
i) a terrible neglect. (Such undue neglect doesn't seem to have helped get the 2009 arXiv successfully peer reviewed either.)
ii) weariness of considering what sort of danger would be presented by taking both very slow evaporation/radiation along with accretion.

Why danger?

Alongside the evaporation, would be the Hawking radiation. One would need only an accretion rate - which would be gradually increasing - of .57kg/s to obtain a radiation level of 5x10^16W - extremely dangerous no doubt even if from within the earth's core. Such an accretion rate and luminosity has been obtained [0808.1415] On the potential catastrophic risk from metastable quantum-black holes produced at particle colliders by using accretion rate formulae given in a CERN risk paper.

I'm not seeing any way that analogous effects could be detectable on compact stars receiving similiar black hole creating collisions from cosmic rays. Also, the accretion rate would become limited so that whole star accretion would not have time to occur.

Note that the Fabi et al 2009 paper is not peer reviewed.

Which one would you (plural) rely on to address safety, concerning the Randall/Sundrum theory that claims to resolve a number of problems in physics;

Casadio/Harms 2002 or Fabi et al 2009?

Smallest stable black-holes

Hypography?

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